在邊点取極大值的函數的漸近積分

<正> 在其中我們設被積分的大數函數係在D域的某種型式的邊界上取絕對極大值。在早先的一篇文章中,作者曾證明了一個關於此類積分的漸近公式,在該處係假定D域的邊界為歐氏空間R_n中的一個(n—1)維曲面。被積分的大數函

ON A KIND OF ASYMPTOTIC INTEGRALS WITH INTEGRANDS HAVING ABSOLUTE MAXIMUM AT BOUNDARY POINTS

It is the purpose of this paper to investigate the asymptotic behavior of a class of integrals of the following form in which f(u)=f(u_1,…, u_n)>0 is assumed to attain an effective absolute maxima at a certain boundary point ξ= (ξ_1,…,ξ_n) of D,D being a simply connected n dimensional closed domain in Euclidean n space.In our investigation, three main types of boundary points ξ's have been distinguished, namely,Type Ⅰ. Let D be an n-dimensional closed domain. We shall call ξ = (ξ_1,…,ξ_n,) a boundary point of type Ⅰ, if it is an ordinary point of the boundary surface S of D and if S has a continuously turning tangent plane near ξ.Type Ⅱ.We shall call ξ = (ξ_1,…,ξ_n) a boundary point of type Ⅱ, if the following conditions are satisfied: (i)ξis a point belonging to the intersection of two (n-1) surfaces S_1 and S_2, where S_1 and S_2 constitutes a part of the boundary surface of D, (ii) ξ is an ordinary point for both S_1 and S_2, where S_1, S_2 have continuously lurning tangent planes near ξ, (iii) the intersection angle θ between tangent planes of S_1 and S_2 at ξ, as measured from the inside of D, is greater than 0 and less than πType Ⅲ. If in the above definition concerning a boundary point of type Ⅱ, the intersection angle θ defined by condition (iii) is equal to zero (i.e. S_1, S_2 have the same tangent plane at ξ), then ξ is called a boundary point of type Ⅲ.In two earlier prapers [1] and [2], the author has studied the case where the function f(u) attains an absolute maxima at a boundary point of type Ⅰ. We are therefore concerned ourselves with the cases for boundary points of types Ⅱ and Ⅲ in the present investigation. Clearly, without loss of generality we may assume a typical boundary point ξ to be origin of the (u)-system, viz.ξ=O=(0,…,0). Moreover, for our need, we assume that arc cos (…) only takes a value between 0 and π. Then our results may be stated as follows:Theorem 1. Let φ(u) and f(u) >0 be defined on D such that1°φ(u) [f(u)]~N is absolutely integrable over D for all N≥0,2°partial derivatives f_i′(u),f_(i,k)″(u) all exist and are continuous. 3°f(u) attains an absolute maxima at a boundary point ξ=(0,…, 0) of type Ⅱ,4°where u→ξ) denotes that u approaches ξ from the inside of D,5°φ(u) is continuous at ξ=(0,…,0) with φ(ξ)≠0. Then for N→∞ we have the asymptotic formula where A =[-ψ_(ik)″(ξ)]is the Hessian matrix of-ψ(u)=-log f(u),α and βare respectively the normal vectors orthogonal to S_1 and to S_2 at ξ such that their intersection angle is equal to that between tangent planes of S_1 and S_2 at ξ, as measured from the inside of D.Clearly, in the particular case n=2, the (n-1) surfaces S_1, S_2 of the above theorem reduce to two boundary curves C_1, C_2 of the plane region D.Theorem 2. Let f(u_1, u_2)>0, φ(u_1, u_2) and D satisfy all the conditions of the above theorem (with n=2) except condition 3° which is replaced by3 f(u_1, u_2) has an absolute maxima at a boundary point ξ=(0, 0) of type Ⅲ. Then for N→∞ we have the formulas according as the positive directions of two principal normals of C_1, C_2 at (0, 0) are just opposite or the same, where R_1, R_2 are respectively the radii of curvatures of C_1, C_2 at (0, 0) andα=-(f(0,0)~(-1){f_(11)"(0,0)cos~2θ+f_12″(0,0)sin2θ+f_(22)″(0,0)sin~2θ},θ being the angle made by the tangent of C_1 (or C_2) at (0, 0) with the u_1-axis.Proof of the first result consists of applying some known results of [1], using some simple matrix algebra and employing the n-dimensional spherical coordinates for the transformation and asymptotic estimation of an n-fold integral. The second result is proved by a method of so-called circle arc-wise integration in which the Frenet-Serret formula is used for an estimation of some arc length.The author has not yet found an explicit asymptotic formula for J(N) with regard to the general case of type Ⅲ.